There is one big mistake in this list. The word which should not be in the list is <s>'"I fancy". </s> Please delete it!
'Fancy' does not mean 'like' - it is not a synonym of the other phrases. You cannot say, for example, "What are your hobbies?" "I fancy music and fashion". This is completely wrong.
The verb 'fancy' is an informal synonym of 'to desire', especially in modern British English. "I fancy xx" means 'I want xx', in the sense that it is attracting you in some way. We use it in two contexts:
<ol><li>To say what you want to do or want to have, especially at a particular moment. For example:</li></ol>
"What do you fancy doing tonight? Fancy a movie?"
"I don't fancy going out this evening. It's too cold".
"Have you looked at the menu? What do you fancy?" "I fancy the smoked salmon. That looks good".
A synonym in many of these cases is 'feel like'.
2. To say whether you "like" a girl or boy in a sexual way.
I have come across this issue with Russian speakers before. Someone, somewhere, decided to put 'fancy' in the list of synonyms for 'like' in a textbook for Russian speakers studying English, and it has been causing problems ever since.
The expression 'keen on' is OK. It mean to be enthusiastic about something, but it's worth pointing out that it's a very British English expression - you're unlikely to meet an American English speaker using it.
The others are fine and all fairly common.